Practical informations

The conference takes place in Brest, France. The city is located by the sea, the air
temperature is between 18 and 28 Celsius degree, the water temperature is between 16 and
18 degree and there are no mosquitos. The beaches are located outside the city and are better
reached by bus/car, a trip is planned on wednesday afternoon.

Direction to the university

Concerning transportation, tickets can be purchased at stations and in buses but not in
tramways. The same ticket works for buses, tramways and telepherique (1 hour duration) and
must be stamped each time you enter a vehicle.

– From the airport

The airport is located North-East of the city. Either take a taxi (15 mn, 30 euros) or take the
shuttle in front of the airport. Shuttle tickets (1.5 euros) are purchased in the shuttle. The
shuttle goes to the tramway terminus station (10 mn) or to the railway station. The tramway
goes to the center of the town (30 mn). Get off at place de la liberté. This is the city center,
from which you can reach the university or your hotel in at most 15 minutes, either by bus or
by foot.

– From the city center

The university can be reached from the town center or the railway station by bus (15mn) or
by foot (30mn). Take the busnumber 1 either in front of the railway station or from place dela liberté. Tickets can be bought on the bus. Wait until the bus takes a huge blue bridge and
get off at Bouguen close to entry A of the faculté des Sciences, just after the roundabout.
A detailed map of the bus stations is available on the website of the Bibus bus
company.

– On the campus

The mathematics department is located on third floor of the bâtiment C, the building closest
to entry A. The secretary is in the library, close to the restroom and the coffee machine. The
talks take place in Amphitheater E.

Titles and abstracts of the talks

Jon Aaronson

In infinite ergodic theory, various weak and distributional limits replace the absolutely
normalized pointwise ergodic theorem. We’ll review the subject and then see that every
random variable on the positive reals occurs as the distributional limit of some infinite
ergodic transformation. As a corollary, we obtain a complete classification of the possible
”A-rational ergodicity properties” for an infinite ergodic transformation.

The main construction follows by ”inversion” from a cutting and stacking construction
showing that every random variable on the positive reals occurs as the distributional limit of
the partial sums of some positive, ergodic stationary process normalized by a 1-regularly
varying normalizing sequence (indeed, here the process can be chosen over any
EPPT).

Sara Brofferio

On unbounded invariant measures of stochastic dynamical systems

We consider stochastic dynamical systems Xn= Ψn(Xn-1), where Ψn are i.i.d. random
continuous transformations of R. We assume that Ψn(x) behave asymptotically like Anx, for
some random positive number An. The main example is the stochastic affine recursion
Xn= AnXn-1+Bn, but this class includes other interesting processes such as reflecting
random walks or branching process. Our aim is to describe invariant Radon measures of the
process {Xn} in the critical case, when ElogA = 0. Under optimal assumptions, we prove
that those measures behave at infinity like dx∕x. In the proof we strongly use some properties
of random walks on the affine group. The talk will be based on a joint paper with Dariusz
Buraczewski.

Michael Bromberg

Temporal distributional limit theorem for cocycles over rotations

For a measure preserving system (X,,μ,T ) and a real valued function f on X,
temporal random variables along an orbit of a fixed point x in X are obtained by
considering the Birkhoff sums Sn(f,x), n = 1,...,N and choosing n randomly uniformly
from 1, ..., N. These r.v’s, measure the fraction of time that Birkhoff sums spend in
various sets. If, under proper normalization, as N tends to infinity, these variables
converge to a non-atomic distribution, we say that f satisfies a temporal limit theorem
along the orbit of x (when the limit is Gaussian, we refer to this as temporal CLT).
The aim of the talk is to introduce the relevant concepts and sketch a proof of a
temporal CLT for piecewise constant cocycles with a single breakpoint, over an
irrational rotation with a badly approximable rotation number. This result generalises
earlier results by J.Beck and by D.Dolgopyat and O.Sarig. This is joint work with
C.Ulcigrai.

Jon Chaika

Ergodicity of typical skew products over some interval exchange transformations

Let T be a linear recurrent interval exchange transformation. This is a measure zero, but full
Hausdorff dimension set of interval exchange transformations that are analogous to badly
approximable rotations. We show that an R valued skew product over such an IET by an
integral 0 function that is a linear combination of characteristic functions of intervals is
typically ergodic. Relevant terms will be defined. This is joint work with Donald
Robertson.

Françoise Dal’bo

An example of a nonuniform lattice with infinite Bowen-Margulis measure

Joint work with M.Peigné, J-C Picaud, A.Sambusetti.I will explain how to construct a noncompact negatively curved Riemannian surface with
finite volume admitting an infinite Bowen-Margulis measure.

Dmitry Dolgopyat

On Local Limit Theorems for hyperbolic flows

I describe an approach to proving local limit theorems and related for flows based on
(multidimensional) local limit theorem for associated Poincare map. Both finite and infinite
measure case will be discussed. Based on a joint work with Peter Nandori.

Rhiannon Dougall

Growth of closed geodesics for infinite covers

We are interested in the dynamics of the geodesic flow for infinite volume manifolds M
which arise as a regular cover of a fixed compact (or convex cocompact) negatively
curved manifold M0. Writing hM for the exponential growth rate of closed geodesics
in M, we have that hM≤ h0, where h0 is the topological entropy of the geodesic
flow for M0. We answer the question of when there is a uniform gap hM< h0 in M
in terms of the permutation representations given by the covering M of M0. The
proof uses the symbolic dynamics for the flow, and so we formulate the analogous
statements for countable state shifts obtained as group extensions of a finite state
shift.

Olivier Glorieux

The aim of my talk will be to explain how classical invariants and theorems for groups acting
on the hyperbolic space, can be extended to the Anti-de Sitter (AdS) setting. We will recall
the notion of critical exponent and Hausdorff dimension for discrete action on the hyperbolic
space and explain how we can define similar notions for a certain type of groups acting on
AdS manifolds. We will finally explain how to get a rigid bound for these invariants in
dimension 3 which is a result equivalent a famous result obtained by R. Bowen in ’79 . This
is a joint work with D. Monclair.

Sébastien Gouëzel

Quantitative Pesin theory for subshifts of finite type

In non-uniformly hyperbolic dynamics, Pesin sets are measurable sets where the
dynamics is very well understood. However, their definition makes these sets hard to
control in a quantitative way, even when the underlying dynamics is hyperbolic. We
will explain why such a control is useful, and what kind of bounds we can obtain.
Joint work with L. Stoyanov.

Alba Málaga Sabogal

Generic Wind-Tree Dynamics

The Wind-Tree is an example of a dynamical system that has a very simple description,
while having a very rich dynamics. It’s a particular case of a billiard: a particle (the wind)
goes straight forward as long as it does not meet any obstacle and it bounces elastically at
each obstacle met. There is an infinite number of square obstacles which are distributed
irregularly all over the plane. The dynamics will strongly depend on the distribution of the
obstacles. The different configurations live in a Baire space - we can thus ask what
happens for a generic configuration (i.e. a configuration in a -dense set). We
found that generic Wind Tree dynamics is actually nice: minimal, ergodic and
of infinite ergodic index in almost every direction. This is joint work with Serge
Troubetzkoy.

Emmanuel Roy

Ergodic splittings of Poisson processes

If N denotes a Poisson process, a splitting of N is formed by two point processes N1 and N2
such that N = N1+N2. If N1 and N2 are independent Poisson processes then the splitting is
said to be Poisson and such a splitting is always available (We allow the possibility to enlarge
the ambient probability space). In general, a splitting is not Poisson but the situation changes
if we require that the distributions of the point processes are invariant by a common
underlying map that acts at the level of each point of the processes. We will prove that if this
map has infinite ergodic index, then a splitting is necessarily Poisson if the environment is
ergodic.

This is a work in progress, with Elise Janvresse and Thierry de la Rue.

Manuel Stadlbauer

Graph extensions of Gibbs-Markov maps and amenability

The aim of the talk is to relate a general notion of amenability of graphs with the probability
of return of a random walk with stationary increments. That is, for a Markov map T : X → X
with embedded Gibbs-Markov structure (i.e. T is a tower over a Gibbs-Markov map with full
branches) and κ a map from X to the automorphisms of the graph, we relate the decay
of

with the amenability of the graph. It turns out that on the level of the embedded
Gibbs-Markov structure, amenability is equivalent to spectral radius equal to 1 of the transfer
operator of the graph extension. In particular, this generalizes results by Kesten, Day and
Derriennic and Guivarc’h for random walks with independent increments. With respect to T ,
the relation is more intricate and requires additional assumptions. These results have
canonical application to the geodesic flow on H∕G, where G is a subgroup of a finitely
generated Fuchsian group.

This is joint work with Johannes Jaerisch (Shimane, Japan) and Elaine Rocha (Salvador,
Brazil).

Dalia Terhesiu

Exploiting semistable laws for i.i.d. random variables

We recall that semistable laws is a class of infinitely divisible laws, which complements the
more well known stable laws. I will recall some main, previously established, results on
necessary and sufficient conditions for the existence of semistable laws for i.i.d. random
variables. I will report on work in progress with Peter Kevei which aims toward a complete
understanding of a limit law for null recurrent renewal chains, assuming that the involved
return function is in the domain of a semistable law (as such, no strict regular variation is
required). Some analogies with the Darling Kac law will be discussed. If time remains, I will
present some results of work in progress with Douglas Coates on semistable laws for interval
intermittent maps.

Damien Thomine

Induction invariance, harmonic functions and applications

Given a random walk on Zd, one may be interested in a large variety of questions on its
statistical porperties (such as ”What is the probability of being at a given site at a given
time?”). Here, I shall discuss questions such as:

- Starting from 0, what is the probability of hitting site p before going back to
0?

- Starting from 0, what is the probability of hitting site p before site q?

In the setting of random walks, the answer to these questions is well known, and involves the
induction invariance of the solutions of the Poisson equation.

In the setting of Zd extensions of dynamical systems, however we lose the Markov
property, and thus we cannot use these tools. However, I’ll show that these can be partially
replaced by a use of Green-Kubo’s formula, which still satisfies some induction
invariance in this more general setting. This gives us answers for dynamical systems
such as the geodesic flow on periodic hyperbolic surfaces, or (in part) Lorentz’
gases.

Joint work with F. Pène (University of Brest).

Roland Zweimüller

Return- and hitting-time distributions of small sets.

I will present some work on the asymptotics of return- and hitting-time distributions of small
sets in certain infinite measure preserving systems, as the measure of these sets decreases to
zero (“rare events”). My focus will be on fairly nice concrete systems and an abstract setup
accommodating them. This includes joint work with F. Pene, B. Saussol, and S.
Rechberger.